Early medieval letters aren't my area of expertise (I work on 13th-15th century literary and intellectual culture), so I would defer to the experience of someone who is more familiar with this field. My answer is based on what I've read for paleography and Latin classes where we've surveyed different textual genres from late antiquity to the early modern period. Also, I can mostly only speak for the British Isles and France.

With those caveats out of the way: I wouldn't be able to put an exact figure on how much early medieval correspondence survives, but there are a good number of fairly extensive collections of letters written to and from prominent churchmen, such as Gregory the Great, Boniface and Alcuin. These individuals sent out a particularly large number of letters because they often maintained geographically large networks of correspondence with other churchmen throughout Europe, often sharing new theological or philosophical ideas with their peers or sending requests for copies of sought-after manuscripts. For example, when Alcuin was living at Charlemagne's court, he wrote a request to his former colleagues asking for a selection of books from the library at York to be sent to him (Letter 121).

A lot of these collections have been translated, although it's not always easy to find them outside of university libraries - the link I'll include at the end has a pretty large bibliography of letter collections, and some of the older ones can be found on archive.org. There's also a corpus of letters with female senders or recipients, which can be found (in translation) here: https://epistolae.ctl.columbia.edu/home. Women's letters tended to be underrepresented in terms of survivals, but it was not unheard of for aristocratic women or those living in nunneries to write and receive letters, including to unrelated men.

We have a lot of surviving letters in the early medieval west because they were often copied down as examples of good letter-writing practice. Since the time of the Roman Empire, letter-writing was considered as a form of rhetorical art, and was commonly taught as part of general education for people entering monastic or clerical careers, and sometimes also for the children of secular rulers. Letters from prominent individuals regarded as having literary merit would be collected in manuscripts for the purpose of this education, or simply for literary enjoyment. As a sidenote to this, very few letters actually survive in the form they were originally sent; as with your example, our earliest copies of many letters were made several centuries after they were originally sent.

In summary: early medieval letters survive in surprisingly high numbers. This figure is higher in western European regions such as England and France - I would be interested myself to know about what factors meant a higher rate of survival in these areas compared to other regions (including your country), but I can't offer anything except speculation on this matter.

This seminar intro and reading list, curated by an expert in early medieval letters, gives a bit more info on the topic, and the exhaustative bibliography can point you to other secondary sources and published letter collections. http://www.lancaster.ac.uk/staff/haywardp/hist424/seminars/03.htm

I would particularly highlight

Poster, C., and L. C. Mitchell (eds), Letter-Writing Manuals and Instruction from Antiquity to the Present: Historical and Bibliographic Studies (Columbia, NY, 2007).

Rosenthal, J. T., ‘Letters and Letter Collections’, in J. T. Rosenthal (ed.), Understanding Medieval Primary Sources: Using Historical Sources to Discover Medieval Europe, Routledge Guides to Using Historical Sources (London, 2012), pp. 72–85.

as good further introductory resources.